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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2016 • Question 5

Question

Math 6420 • PDE Fall 2016 • Question 5

Let \(T:X\to X\) be a bounded linear operator on a complex Banach space \(X\). Prove that \(\sigma (T)\) lies in the complex plane disk: \[ \{\lambda \in \mathbb {C}\mid \lvert \lambda \rvert \le \lVert T\rVert \}. \]

Raw LaTeX

Let \(T:X\to X\) be a bounded linear operator on a complex Banach space \(X\). Prove that
\(\sigma(T)\) lies in the complex plane disk:
\[
  \{\lambda\in\mathbb{C}\mid \lvert\lambda\rvert\le\lVert T\rVert\}.
\]

Topic Tags

Banach and Normed Spaces Bounded Linear Operators Dual Spaces and Bounded Linear Functionals Linear Programming and Optimization